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A. V. Andreev, "Limiting equilibrium analysis of curvilinear boundary cracks in an elastic half-plane, with the stress asymptotics taken into account," Mech. Solids. 38 (6), 65-77 (2003)
Year 2003 Volume 38 Number 6 Pages 65-77
Title Limiting equilibrium analysis of curvilinear boundary cracks in an elastic half-plane, with the stress asymptotics taken into account
Author(s) A. V. Andreev (Moscow)
Abstract It is frequently the case for fracture processes that the stress state near a crack is equivalent to a field of compression and shear. In such situations, the problems of fracture mechanics can be stated as a combination of a contact problem and a problem of mechanics of a deformable solid. These problems require special methods for their solution, because the distribution of the contact regions is unknown. Contact between the crack edges can be caused not only by external loads, but may be due to the shape of the crack [1, 2], or the interaction between cracks and other defects [3, 4], or their interaction with the boundary of the elastic body [1, 5]. Thus, an investigation of the limiting equilibrium of boundary cracks should be performed with the possibility of contact between their surfaces taken into account.

In this work, we consider a two-dimensional problem for the stress-strain state of an elastic half-plane weakened by a curvilinear surface crack whose surfaces are interacting with friction. A method for the calculation of the limiting equilibrium of such cracks is developed on the basis of the solution of a system of singular integral equations corresponding to a mixed (contact) boundary-value problem of elasticity. In this investigation, we take into consideration the behavior of the solution near the vertex on the boundary, and for this purpose an approach utilizing the results of previous asymptotic analysis [5] is proposed. In the present paper, we focus on taking into account, in a correct manner, the behavior of the solution near the surface crack. It is shown that its asymptotic behavior near the boundary substantially affects the limiting equilibrium of cracks. Solutions are obtained for problems of equilibrium of cracks with no interaction between their surfaces, as well as for cracks whose surfaces are in partial contact.

Surface cracks in an elastic half-plane with no contact between their surfaces were considered, for example, in [1, 6-8]. Quasistatic growth of an arbitrary curvilinear surface crack in an elastic half-plane was studied in [9], with possible contact between its surfaces being neglected. In [10], a similar problem was formulated with possible surface contact taken into account. Two-dimensional (three-dimensional) problems for rectilinear (plane) surface cracks whose surfaces are in contact with friction were studied in [8], and some kinked cracks in an elastic half-plane with surfaces interacting with friction were considered in [11]. In [12], a problem of a rigid punch acting on an elastic half-plane with curvilinear cracks is studied with possible contact of their surfaces taken into account (without friction or with complete adhesion in the case of contact). In [5], a method is proposed for the investigation of surface cracks in two-dimensional domains, with friction between their interacting surfaces taken into account. Here, this method is refined to take into account quantitative characteristics of the asymptotic behavior of the solution near the boundary.
References
1.  M. P. Savruk, Two-Dimensional Elasticity Problems for Bodies with Cracks [in Russian], Naukova Dumka, Kiev, 1981.
2.  A. V. Andreev, R. V. Goldstein, and Yu. V. Zhitnikov, "Equilibrium of curvilinear cuts, with the formation of regions of overlapping, sliding, and adhesion on the crack edges taken into account," Izv. AN. MTT [Mechanics of Solids], No. 3, pp. 137-148, 2000.
3.  V. Tamuzs, V. Petrova, and N. Romalis, "Plane problem of macro-microcrack interaction taking account of crack closure," Eng. Fract. Mech., Vol. 55, No. 6, pp. 957-967, 1996.
4.  M. Comninou and F.-K. Chang, "Effects of closure and friction on a radial crack emanating from a circular hole," Intern. J. Fracture, Vol. 28, No. 1, pp. 29-36, 1985.
5.  A. V. Andreev, R. V. Goldstein, and Yu. V. Zhitnikov, Calculation of the Limiting Equilibrium of Internal and Boundary Cracks with Interacting Surfaces in an Elastic Half-plane. Preprint No. 692 [in Russian], In-t Problem Mekhaniki RAN, Moscow, 2001.
6.  V. V. Panasyuk, M. P. Svaruk, and A. P. Datsyshin, Stress Distribution near Cracks in Plates and Shells [in Russian], Naukova Dumka, Kiev, 1976.
7.  W. Zang and P. Gudmundson, "An integral equation model for piecewise smooth cracks on an elastic half-plane," Eng. Fract. Mech., Vol. 32, No. 6, pp. 889-897, 1989.
8.  N.-A. Noda, M. Yagishita, and T. Kihara, "Effect of crack shape, inclination angle, and friction coefficient in crack surface contact problem," Intern. J. Fracture, Vol. 105, No. 4, pp. 367-389, 2000.
9.  A. P. Datsyshin, G. P. Marchenko, and V. V. Panasyuk, "On the theory of crack growth in the case of rolling contact," Fiz.-Khim. Mekh. Mater., Vol. 29, No. 4, pp. 49-61, 1993.
10.  O. P. Datsyshyn and V. V. Panasyuk, "Durability and fracture calculation model of solids under their contact interaction," in Proc. 11th Biennial Europ. Conf. Fracture. Volume 2, pp. 1381-1386, France, 1996.
11.  W. Zang and P. Gudmundson, "Frictional contact problems of kinked cracks modelled by a boundary integral method," Intern. J. Numer. Mech. in Eng., Vol. 31, No. 3, pp. 427-446, 1991.
12.  V. V. Panasyuk, O. P. Datsyshyn, and G. P. Marchenko, "Stress state of a half-plane with cracks under rigid punch action," Intern. J. Fracture, Vol. 101, No. 4, pp. 347-363, 2000.
13.  N. I. Muskhelishvili, Some Basic Problems in Mathematical Elasticity [in Russian], Nauka, Moscow, 1966.
14.  N. I. Muskhelishvili, Singular Integral Equations [in Russian], Nauka, Moscow, 1968.
15.  F. E. Erdogan, G. D. Gupta, and N. S. Cook, "The numerical solutions of singular integral equations," in Mechanics of Fracture. Volume 1, pp. 368-425, Noordhoff Intern. Publ., Leyden, 1973.
16.  A. M. Lin'kov, The Complex Method of Boundary Integral Equations of Elasticity [in Russian], Nauka, St. Petersburg, 1999.
17.  M. M. Chawla and T. R. Ramacrishnan, "Modified Gauss-Jacobi quadrature formula for the numerical evaluation of Cauchy type singular integrals," BIT, Vol. 14, No. 1, pp. 14-21, 1974.
18.  A. I. Kalandiya, Mathematical Methods of Two-Dimensional Elasticity [in Russian], Nauka, Moscow, 1973.
19.  A. V. Andreev, Calculation of Limiting Equilibrium of Boundary Curvilinear Cracks in an Elastic Half-plane, with Stress Singularities Taken into Account. Preprint No. 730 [in Russian], In-t Problem Mekhaniki RAN, Moscow, 2003.
20.  M. L. Williams, "Stress singularities resulting from various boundary conditions in angular corners of plates in extension," J. Appl. Mech., Vol. 19, No. 4, pp. 526-528, 1952.
21.  R. V. Goldstein and Yu. V. Zhitnikov, "Analysis of the process of sliding of crack surfaces, with friction forces under complex loading taken into account," Izv. AN. MTT [Mechanics of Solids], No. 1, pp. 139-148, 1991.
Received 25 April 2002
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